179ISSN 1746-7659, England, UK Journal of Information and Computing Science

Vol. 2, No. 3, 2007, pp. 179-190

Estimating Error Bounds for Non-stationary Binary Subdivision Curves / surfaces *

Ghulam Mustafa +, Sadiq Hashmi and Faheem Khan

Department of Mathematics, The Islamia University of Bahawal pur Pakistan

( Received 23 August 2006, accepted 5 October 2006 )

Abstract. Error bounds between non-stationary binary subdivision curves/surfaces and their control polygons after k-fold subdivision are estimated. The bounds can be evaluated without recursive subdivision. A computational formula of subdivision depth for non-stationary binary subdivision surfaces is also presented. From this formula one can predict the subdivision depth within a user specified error tolerance. Our results not only remove errors in the results of [3] but also contain the generalization of their results.

Keywords: Subdivision curve, subdivision surface, subdivision depth, error bound.

1. Introduction In recent years subdivision schemes have been important because they provide an efficient way to

describe curves, surfaces and other geometric objects. Subdivision allows to generate smooth geometric objects from a given control polygon. With this coarse polygon a sequence of refined polygon can be computed. In the limit this sequence of polygon converges to a continuous smooth object. Each refinement step can be divided into two different aspects. First, a topological operation is performed. Therefore new vertices are added to the control polygon and the rectangles/triangles are split. Then the geometry of the control polygon is changed by a smoothing operation. The question is how refined polygon approximates the limiting curve/surface [1], [2], [3], [5], [6], [7], [8], [9] and [10]. This question appears in many applications as rendering, intersection, testing and design. Existing methods only compute the error bounds for stationary subdivision schemes. To best of our knowledge no work has been done to compute the error bounds for non-stationary subdivision schemes. As non-stationary subdivision schemes are the generalization of stationary subdivision schemes. Therefore we are interested in getting error bounds estimations between non-stationary subdivision curves/surfaces and their control polygons. In this article, for simplicity we have used the notations and methodology of [3].

The rest of the paper is arranged as follows. In Section 2, we give the definition of non-stationary subdivision curves and gather some notations to set out our terminology needed in Section 3. We present our main result to estimate error bounds between non-stationary subdivision curve and its control polygon in Section 3. In Section 4, we give the definition of non-stationary subdivision surfaces and gather some new notations to set out our terminology needed in Section 5. We present our main result to estimate error bounds between non-stationary subdivision surface and its control polygon in Section 5. In this Section we also derive a computational formula of subdivision depth for non-stationary subdivision surfaces.

2. Non-stationary subdivision curves Given a set of control points , Nk

i Rp ∈ 2, ≥∈ NZi , where k is a nonnegative integer. A non-stationary binary subdivision scheme is defined by

(2.1)

⎪⎪⎩

⎪⎪⎨

⎧

=

=

∑

∑

=+

++

=+

+

m

j

kji

kj

ki

m

j

kji

kj

ki

pbp

pap

0

112

0,

12

,

* This work is supported by the Indigenous Ph. D Scholarship Scheme of Higher Education Commission (HEC) Pakistan. + Corresponding author.mustafa_rakib@yahoo.com. Contact address: sadiqhashmi@iub.edu.pk, fahimscholar@gmail.com

Published by World Academic Press, World Academic Union

G. Mustafa, et al: Estimating error bounds for non-stationary binary subdivision curves/surfaces 180

where is greater than zero. The coefficients m { }m

jkja

0= and { }m

jkjb

0=are called the mask at the kth level of the

subdivision scheme. If the mask is independent of k , then the scheme is called stationary, otherwise it is called non-stationary. A necessary condition for the uniform convergence of the subdivision scheme (2.1) on the diadic points for arbitrary initial data, is that

.100

≈≈ ∑∑==

m

j

kj

m

j

kj ba (2.2)

2.1. Notations We gather here some notations to set out our terminology needed in the sequel.

⎪⎩

⎪⎨

⎧

+−=

−=

+++

+

.)(21max

,max

1112

2

12

1

ki

ki

kiik

ki

kiik

pppT

ppT (2.3)

,,max1

0

1

0 ⎭⎬⎫

⎩⎨⎧

= ∑∑−

=

−

=

m

j

kj

m

j

kj BAφ (2.4)

where

.,1,,11

∑∑+=+=

≥==m

ji

ki

kj

m

ji

ki

kj jbBaA .

21

10 −∑

=

m

i

ki

k bB

{ }DC ˆ,ˆmax=ϕ (2.5) where

,,...,2,1,maxˆ1

0

1

⎭⎬⎫

⎩⎨⎧

== ∑−

=

−m

j

lj klCC ,,...,2,1,maxˆ

0

1

⎭⎬⎫

⎩⎨⎧

== ∑=

−m

j

lj klDD ∑

=

−=j

i

ki

ki

kj baC

0)( and . k

jkj

kj CaD −=

3. The error bounds for non-stationary subdivision curves The aim of this section is to estimate error bounds between limiting non-stationary subdivision curves

and their control polygons. First we prove the following Lemma’s needed for Theorem 3.5.

Lemma 3.1. Given initial control polygon ,0ii pp = Zi∈ , let the values be defined

recursively by non-stationary subdivision scheme (2.1) together with (2.2) then ,k

ip 0≥k

ki

kii

m

j

kjk ppAT −≤ +

−

=∑ 1

1

0

1 max , (3.1)

where and are defined in (2.3) and (2.4) respectively. 1kT k

jA

Proof: From (2.1) and (2.2)

∑ ∑= =

++

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

m

j

ki

m

j

kj

kji

kj

ki

ki papapp

0 0

12 .

This implies

( )( ) ( )( ) ......... 12321211

2 +−++++−+++=− ++++ k

iki

km

kkki

ki

km

kkki

ki ppaaappaaapp

+ ( )( ) ( )( )kmi

kmi

km

kmi

kmi

km

km ppappaa 1211 −++−+−+− −+−+ .

From this we get

( )∑−

=+++

+ −=−1

01

12

m

j

kji

kji

kj

ki

ki ppApp ,

where If ∑+=

=m

ji

ki

kj aA

1. k

ikiik ppT −= +1

21 max , then k

ikii

m

j

kjk ppAT −≤ +

−

=∑ 1

1

0

1 max . This completes the proof.

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Journal of Information and Computing Science, 2 (2007) 3, pp 179-190 181

Lemma 3.2. Given initial control polygon ,0ii pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (2.1) together with (2.2) then ,k

ip 0≥k

ki

kii

m

j

kjk ppBT −≤ +

−

=∑ 1

1

0

2 max ,

where and are defined in (2.3) and (2.4) respectively. 2kT k

jB

Proof: From (2.1) and (2.2)

( )∑ ∑=

+=

++++ +⎟⎟

⎠

⎞⎜⎜⎝

⎛−=+−

m

j

ki

ki

m

j

kj

kji

kj

ki

ki

ki ppbpbppp

01

01

112 2

1)(21

.

( )ki

ki

kji

kji

m

j

kj ppppb 1

021

+++=

−−+⎟⎠⎞

⎜⎝⎛= ∑ ,

( ){ ( )ki

ki

kki

ki

kki

ki

ki ppbppbppp −+−−=+− +++++ 11101112 2

1)(21

( ) ( )}ki

ki

ki

kmi

kmi

kmi

km

ki

ki

ki

ki

k ppppppbppppb −+−+−++−+−+ ++−+−+++++ 11111122 ...... ,

( )( ) ( )( )ki

ki

km

kkki

ki

km

kkki

ki

ki ppbbbppbbbppp 12321101

112 ......

21)(

21

++++++ −++++−+++−=+−

( )( ) ( )( ) ( )( )kmi

kmi

km

kmi

kmi

km

km

ki

ki

km

kk ppbppbbppbbb 12112343 ...... −++−+−+−++ −+−+++−++++ .

Since from (2.2), , then we have km

i

ki bb 0

11 −=−∑

=

( )∑−

=++++

++ −=+−

1

011

112 )(

21 m

j

kji

kji

kj

ki

ki

ki ppBppp ,

where , ∑+=

=m

ji

ki

kj bB

1∑=

−=≥m

i

ki

k bBj1

0 21,1 .

If

)(21max 1

112

2 ki

ki

kiik pppT +++ +−= ,

then

ki

kii

m

j

kjk ppBT −≤ +

−

=∑ 1

1

0

2 max ,

This completes the proof.

Lemma 3.3. Given initial control polygon ,0ii pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (2.1) together with necessary conditions (2.2). Suppose ,k

ip 0≥k

kP be the piecewise linear interpolation to the values then ,kip

ki

kii

kk ppPP −≤− +∞

+1

1 maxφ , (3.3)

where φ is defined in (2.4).

Proof: Let ∞

. denote the uniform norm. Since the maximum difference between 1+kP and kP is

attained at a point on the ( )thk 1+ mesh, then

{ }211 ,max kkkk TTPP ≤−

∞

+ , (3.4)

where

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G. Mustafa, et al: Estimating error bounds for non-stationary binary subdivision curves/surfaces 182

⎪⎩

⎪⎨

⎧

+−=

−=

+++

+

.)(21max

,max

1112

2

12

1

ki

ki

kiik

ki

kiik

pppT

ppT (3.5)

If

⎭⎬⎫

⎩⎨⎧

= ∑∑−

=

−

=

1

0

1

0,max

m

j

kj

m

j

kj BAφ ,

then from (3.1), (3.2), (3.4) and (3.5) we have

ki

kii

kk ppPP −≤− +∞

+1

1 maxφ .

This completes the proof.

Lemma 3.4. Given initial control polygon ,0ii pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (2.1) together with (2.2), if ,k

ip 0≥k1<ϕ , then

( ) 0011 maxmax iii

kki

kii

pppp −≤− ++ ϕ , (3.6)

where ϕ is defined in (2.5).

Proof: We claim:

, (3.7) (∑−

=

−+

−++

−+ −=−

1

0

111

1212

m

j

kji

kji

kj

ki

ki ppCpp )

where . ∑=

−−− −=j

i

ki

ki

kj baC

0

111 )(

We can prove above claim by induction on . Let m 1=m then from (2.1)

( ) ( ) 11

11

11

110

10212

−+

−−−−−+ −+−=− k

ikkk

ikkk

iki pabpabpp .

From (2.2) we have . This implies 1

01

01

11

1−−−− −=− kkkk baab

( )∑−

=

−+

−++

−+ −=−

11

0

111

1212 ,

j

kji

kji

kj

ki

ki ppCpp

where . This proves our claim for∑=

−−− −=j

i

ki

ki

kj baC

0

111 )( 1=m .

Now we prove our claim for . From (2.1) 2=m( ) ( ) ( ) 1

21

21

21

11

11

111

01

0212−+

−−−+

−−−−−+ −+−+−=− k

ikkk

ikkk

ikkk

iki pabpabpabpp .

From (2.2) we have . This implies 11

11

10

10

12

12

−−−−−− −+−=− kkkkkk babaab

( )( ) ( )( )11

12

11

11

10

10

111

10

10212

−+

−+

−−−−−−+

−−+ −−+−++−=− k

iki

kkkkki

ki

kkki

ki ppbabappbapp .

Thus we get

( )∑−

=

−+

−++

−+ −=−

12

0

111

1212

j

kji

kji

kj

ki

ki ppCpp ,

where . This proves our claim for ∑=

−−− −=j

i

ki

ki

kj baC

0

111 )( 2=m . Similarly we can find that claim is true

for all m . We also claim:

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Journal of Information and Computing Science, 2 (2007) 3, pp 179-190 183

) , (3.8) (∑=

−+

−++

−++ −=−

m

j

kji

kji

kj

ki

ki ppDpp

0

111

11222

where .)( 11

0

1111 −−

=

−−−− −=+−=∑ kj

kj

j

i

kj

ki

ki

kj CaaabD

Similarly we can prove claim by induction on . Let m 1=m then from (2.1)

( ) 12

11

11

11

10

1101222

−+

−−+

−−−−++ +−+−=− k

ikk

ikkk

ikk

iki papbapbpp .

Since from (2.2) , then we have 10

10

11

11

−−−− +−= kkkk baba

( ){ }( )+−+−=− −−+

−−−++

111

10

10

101222

ki

ki

kkkki

ki ppaabpp

( ) ( ){ }( )11

12

11

11

11

10

10

−+

−+

−−−−− −+−+− ki

ki

kkkkk ppaabab . This implies

( )∑=

−+

−++

−++ −=−

1

0

111

11222

j

kji

kji

kj

ki

ki ppDpp ,

where This proves our claim for . Similarly we can

prove the claim for all m .

.)( 11

0

1111 −−

=

−−−− −=+−=∑ kj

kj

j

i

kj

ki

ki

kj CaaabD 1=m

From (3.7) and (3.8) we get

111

1

0

11 maxmax −−

+

−

=

−+ −⎟⎟

⎠

⎞⎜⎜⎝

⎛≤− ∑ k

ikii

m

j

kj

ki

kii

ppCpp , 111

0

11 maxmax −−

+=

−+ −⎟⎟

⎠

⎞⎜⎜⎝

⎛≤− ∑ k

ikii

m

j

kj

ki

kii

ppDpp .

Recursively we get

( ) 0011 maxˆmax iii

kki

kii

ppCpp −≤− ++ , ( ) 0011 maxˆmax iii

kki

kii

ppDpp −≤− ++ ,

where

⎭⎬⎫

⎩⎨⎧

= ∑ ∑ ∑−

=

−

=

−

=

−−1

0

1

0

1

0

021 ,...,,maxˆm

j

m

j

m

jj

kj

kj CCCC ,

and

⎭⎬⎫

⎩⎨⎧

= ∑ ∑ ∑= = =

−−m

j

m

j

m

jj

kj

kj DDDD

0 0 0

021 ,...,,maxˆ .

If { }DC ˆ,ˆmax=ϕ then

( ) 0011 maxmax iii

kki

kii

pppp −≤− ++ ϕ .

This completes the proof. Remarks 3.1. The main problem in Theorem 1 [3] is that its proof contains error on page 599 in

equation (9). Its corrected form is given in Equation (3.7) of our Lemma 3.4.

Theorem 3.5. Given initial control polygon ,0ii pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (2.1) together with necessary conditions (2.2). Suppose ,k

ip 0≥k

kP be the piecewise linear interpolation to the values and kip ∞P be the limit curve of the scheme (2.1). If

1<ϕ , (3.9) then error bounds between limit curve and its control polygon after k-fold subdivision are

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

≤−∞

∞

ϕϕφβ

1

kk PP , (3.10)

where 001max iii

pp −= +β , φ and ϕ are defined in (2.4) and (2.5).

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G. Mustafa, et al: Estimating error bounds for non-stationary binary subdivision curves/surfaces 184

Proof: From (3.3) and (3.6) we have ( ) 001

1 max iii

kkk ppPP −≤− +∞

+ ϕφ . If 001max iii

pp −= +β

then ( )kkk PP ϕφβ≤−∞

+1 . Using triangle inequality we have ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−

≤−∞

∞

ϕϕφβ

1

kk PP . This completes

the proof. Remarks 3.2/ In non-stationary subdivision schemes the subdivision mask is updated during each

subdivision level. In particular, if the subdivision mask does't change by increasing subdivision level then non-stationary schemes coincides with stationary schemes. In this particular case, Theorem 3.5 is exactly the same with Theorem 1(after correction) of Mustafa et al. [3].

Corollary 3.6. Given initial control polygon ,0ii pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme for curve interpolation [4]. Suppose

,kip 0≥k

kP be the piecewise linear interpolation to the values and k

ip ∞P be the limit curve of the subdivision scheme. Then

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

≤−∞

∞

ϕϕφβ

1

kk PP ,

where 001max iii

pp −= +β , 000 221,

21

21,1max www +=

⎭⎬⎫

⎩⎨⎧

+++= ϕφ and is initial weight

parameter.

ww =0

Proof: A non-stationary subdivision scheme for curve interpolation [4] have following subdivision mask

( ) ( )0,0,1,0,,, 3210 =kkkk aaaa , ( ) ⎟⎠⎞

⎜⎝⎛ −++−= kkkkkkkk wwwwbbbb ,

21,

21,,,, 3210 .

Since above subdivision mask approximately satisfy (3.9) for , then by (3.10) we have the result.

≤≤− kw2499.0 2499.0

4. Non-stationary subdivision surfaces Given a set of control points , Nk

ji Rp ∈, 2,, ≥∈ NZji , where k is a nonnegative integer. A tensor product of non-stationary binary subdivision scheme (2.1) is defined by

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

=

=

=

∑∑

∑∑

∑∑

∑∑

= =++

+++

= =++

++

= =++

++

= =++

+

m

r

m

s

ksjri

ks

kr

kji

m

r

m

s

ksjri

ks

kr

kji

m

r

m

s

ksjri

ks

kr

kji

m

r

m

s

ksjri

ks

kr

kji

pbbp

pabp

pbap

paap

0 0,

112,12

0 0,

12,12

0 0,

112,2

0 0,

12,2

(4.1)

where is greater than zero. The coefficients m { }m

jkja

0= and { }m

jkjb

0= are called the mask at the kth level of

the subdivision scheme. If the mask is independent of , then the scheme is called stationary, otherwise it is called non-stationary. A necessary condition for the convergence of the subdivision scheme (4.1) for arbitrary initial data, is that

k

.100

≈≈ ∑∑==

m

j

kj

m

j

kj ba (4.2)

4.1. Notations We again gather here some new notations to set out our terminology needed in the coming Section.

{ }CBDACA ˆˆ,ˆˆ,ˆˆmax=ψ , (4.3)

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Journal of Information and Computing Science, 2 (2007) 3, pp 179-190 185 where

⎭⎬⎫

⎩⎨⎧

== ∑=

−m

r

jr kjaA

0

1 ,...,2,1,maxˆ ,⎭⎬⎫

⎩⎨⎧

== ∑=

−m

r

jr kjbB

0

1 ,...,2,1,maxˆ ,

⎭⎬⎫

⎩⎨⎧

== ∑−

=

−1

0

1 ,...,2,1,maxˆm

r

jr kjCC ,

⎭⎬⎫

⎩⎨⎧

== ∑=

−m

r

jr kjDD

0

1 ,...,2,1,maxˆ ,

∑=

−=r

t

kt

kt

kr baC

0)( and . k

rkr

kr CaD −=

⎪⎪⎩

⎪⎪⎨

⎧

+⎟⎠

⎞⎜⎝

⎛+=⎟

⎠

⎞⎜⎝

⎛+=

+⎟⎠

⎞⎜⎝

⎛+=⎟

⎠

⎞⎜⎝

⎛+=

∑∑∑∑

∑∑∑∑−

==

−

==

−

==

−

==

,41,

,21,

1

1104

1

1103

1

1102

1

1101

m

s

ks

m

t

kt

km

s

ks

m

t

kt

k

m

s

ks

m

t

kt

km

s

ks

m

t

kt

k

BbbAab

BbaAaa

ηη

ηη (4.4)

where and , ∑+=

=m

st

kt

ks aA

1∑

+=

=m

st

kt

ks bB

1

⎪⎪⎩

⎪⎪⎨

⎧

++=++=

+=+=

∑∑∑∑∑∑

∑∑∑∑∑∑−

===

−

===

−

===

−

===

,21,

21

,,1

1014

1

1013

1

1012

1

1011

m

s

ks

m

r

kr

m

t

kt

m

s

ks

m

r

kr

m

t

kt

m

s

ks

m

r

kr

m

t

kt

m

s

ks

m

r

kr

m

t

kt

BbbBab

AbaAaa

ττ

ττ (4.5)

and

⎪⎪⎩

⎪⎪⎨

⎧

+⎟⎠

⎞⎜⎝

⎛+=⎟

⎠

⎞⎜⎝

⎛+=

⎟⎠

⎞⎜⎝

⎛+=⎟

⎠

⎞⎜⎝

⎛+=

∑∑∑∑∑∑

∑∑∑∑∑∑−

===

−

===

−

===

−

===

.41,

,,

1

1114

1

1113

1

1112

1

1111

m

s

ks

m

t

kt

m

t

kt

m

s

ks

m

t

kt

m

t

kt

m

s

ks

m

t

kt

m

t

kt

m

s

ks

m

t

kt

m

t

kt

BbbAab

BbaAaa

ξξ

ξξ (4.6)

( )

( )

( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

+++−=

+−=

+−=

−=

+++++

++

++

+

+++

+

.41max

,21max

,21max

,max

1,11,,1,1

12,12,

4

1,,1

12,2,

3

,1,1

2,12,

2

,12,2,

1

kji

kji

kji

kji

kjijik

kji

kji

kjijik

kji

kji

kjijik

kji

kjijik

pppppM

pppM

pppM

ppM

(4.7)

⎪⎪⎩

⎪⎪⎨

⎧

−=Δ

=Δ=−=Δ−=Δ

+++

+

+

,

.3,2,1,max,,

1,1,13,,

,,,,1,2,,

,,11,,

kji

kji

kji

ktjiji

kt

kji

kji

kji

kji

kji

kji

pp

tpppp

γ (4.8)

5. The error bounds for non-stationary subdivision surfaces The aim of this section is to estimate error bounds between limiting non-stationary subdivision surfaces

and their control polygons. First we prove the following Lemmas needed for Theorem 5.7.

Lemma 5.1. Given initial control polygon ,,0, jiji pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (4.1) together with (4.2), then ,,

kjip 0≥k

kkkkM 3121111 γξγτγη ++≤ , (5.1)

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G. Mustafa, et al: Estimating error bounds for non-stationary binary subdivision curves/surfaces 186

where and are defined in (4.4)-(4.8). 1111 ,,, kMξτη ,3,2,1, =tk

tγProof: From (4.1) and (4.2) we get

(∑ ∑= =

+++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

r

m

s

kji

ksjri

ks

kr

kji

kji ppaapp

0 0,,,

12,2 )

)

)

)

)

)

. (5.2)

Since

( ) ( ) ( )( )( )( )....

...

,1,1,2,2,3,1,,

,1,1,2,2,3,3

,1,1,2,2

,1,1,,00

,,

kji

kjri

kjri

kjri

kjri

kjri

kmjri

kmjri

km

kji

kjri

kjri

kjri

kjri

kjri

k

kji

kjri

kjri

kjri

k

kji

kjri

kkji

kjri

km

s

kji

ksjri

ks

ppppppppa

ppppppa

ppppa

ppappappa

−+−+−++−

++−+−+−

+−+−

+−+−=−

++++++++++−++++

++++++++++

++++++

+++=

++∑

Hence

( ) ( )

( ) ( ,1

1,1,

1,1,

,,00

,,

∑∑

∑−

=+++++

=++

+=

++

−+−

+−=−

m

s

ksjri

ksjri

ks

m

t

kji

kjri

kt

kji

kjri

km

s

kji

ksjri

ks

ppAppa

ppappa

where . Taking sum on both sides of above equation we get ∑+=

=m

st

ki

ks aA

1

(5.3)

( ) ( )

( ) ( .1

1,1,

01,1,

0

,,0

00

,,0

⎟⎠

⎞⎜⎝

⎛−+⎟

⎠

⎞⎜⎝

⎛−

+−=⎟⎠

⎞⎜⎝

⎛−

∑∑∑ ∑

∑∑∑−

=+++++

==++

=

+==

++=

m

s

ksjri

ksjri

ks

m

r

kr

m

t

kji

kjri

m

r

kr

kt

kji

kjri

m

r

kr

km

s

kji

ksjri

ks

m

r

kr

ppAappaa

ppaappaa

Since

( ) ( ) (( )

( )....

...

,,1,1,2,2,3,1,

,,1,1,2,2,33

,,1,1,22,,110

,,

kji

kji

kji

kji

kji

kji

kjmi

kjmi

km

kji

kji

kji

kji

kji

kji

k

kji

kji

kji

kji

kkji

kji

km

r

kji

kjri

kr

ppppppppa

ppppppa

ppppappappa

−+−+−++−

++−+−+−+

−+−+−=−

+++++−++

+++++

++++=

+∑

Hence

( ) ( ) (∑∑∑−

=+++

=+

=+ −+−=−

1

1,,1

1,,1

0,,

m

s

kjsi

kjsi

ks

m

t

kji

kji

kt

m

r

kji

kjri

kr ppAppappa .

Similarly

( ) ( )

( ) ( .1

11,1,1

1,1,1

,1,00

,1,

∑∑

∑−

=+++++

=++

+=

++

−+−

+−=−

m

s

kjsi

kjsi

ks

m

t

kji

kji

kt

kji

kji

km

r

kji

kjri

kr

ppAppa

ppappa

Substituting these sum into (5.3) and then from (5.2) we have

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Journal of Information and Computing Science, 2 (2007) 3, pp 179-190 187

)

( ) ( )

( ) ( )

( ) ( .,1,

1

101,1,1

1

11

,,1

1

10,1,

10

1,1,1

2

1,,1

10,

12,2

⎟⎠

⎞⎜⎝

⎛−+−⎟

⎠

⎞⎜⎝

⎛

+−+−⎟⎠

⎞⎜⎝

⎛

+−⎟⎠

⎞⎜⎝

⎛+−⎟

⎠

⎞⎜⎝

⎛=−

+++++

−

==+++++

−

==

+++

−

=+

=

+++=

+=

+

∑∑∑∑

∑∑

∑∑

ksjri

ksjri

m

s

ks

m

r

kr

kjsi

kjsi

m

s

ks

m

t

kt

kjsi

kjsi

m

s

ks

kkji

kji

m

t

kt

k

kji

kji

m

t

kt

kji

kji

m

t

kt

kkji

kji

ppAappAa

ppAappaa

ppappaapp

This implies

,max

max

max

3,,,

1

111

2,,,

1

101

1,,,

1

110

1

kjiji

m

s

ks

m

t

kt

m

t

kt

kjiji

m

s

ks

m

r

kr

m

t

kt

kjiji

m

s

ks

m

t

kt

kk

Aaa

Aaa

AaaM

Δ⎟⎠

⎞⎜⎝

⎛++

Δ⎟⎠

⎞⎜⎝

⎛++

Δ⎟⎠

⎞⎜⎝

⎛+≤

∑∑∑

∑∑∑

∑∑

−

===

−

===

−

==

where kji

kjijik ppM ,

12,2,

1 max −= + .

Using notations from (4.4)-(4.8) we can rewrite above inequality . This completes the proof.

kkkkM 3121111 γξγτγη ++≤

By using the technique of Mustafa et al. [3] and the one used in Lemma 5.1 one can easily prove the following Lemmas.

Lemma 5.2. Given initial control polygon ,,0, jiji pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (4.1) together with (4.2), then ,,

kjip 0≥k

, (5.4) kkkkM 3222122 γξγτγη ++≤

where and are defined in (4.4)-(4.8). 2222 ,,, kMξτη ,3,2,1, =tk

tγ

Lemma 5.3. Given initial control polygon ,,0, jiji pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (4.1) together with (4.2), then ,,

kjip 0≥k

kkkkM 3323133 γξγτγη ++≤ , (5.5)

where and are defined in (4.4)-(4.8). 3333 ,,, kMξτη ,3,2,1, =tk

tγ

Lemma 5.4. Given initial control polygon ,,0, jiji pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (4.1) together with (4.2), then ,,

kjip 0≥k

kkkkM 3424144 γξγτγη ++≤ , (5.6)

where and are defined in (4.4)-(4.8). 4444 ,,, kMξτη ,3,2,1, =tk

tγ

Lemma 5.5. Given initial control polygon ,,0, jiji pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (4.1) together with (4.2). Suppose

,,k

jip 0≥kkP be the piecewise

linear interpolation to the values then ,,k

jipkkkkk PP 321

1 ξγτγηγ ++≤−∞

+ , (5.7)

where { }4321 ,,,max ηηηηη = , { }4321 ,,,max τττττ = and { }4321 ,,,max ξξξξξ = , where

4...1,,, =tttt ξτη are defined in (4.4)-(4.6), are defined in (4.8). ,3,2,1, =tktγ

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G. Mustafa, et al: Estimating error bounds for non-stationary binary subdivision curves/surfaces 188

Proof: Let ∞

. denote the uniform norm. Since the maximum difference between 1+kP and kP is

attained at a point on the ( )thk 1+ mesh, then

{ }43211 ,,,max kkkkkk MMMMPP ≤−

∞

+ , (5.8)

where are defined in (4.7). 4321 ,,, kkkk MMMM

If { }4321 ,,,max ηηηηη = , { }4321 ,,,max τττττ = and { }4321 ,,,max ξξξξξ = , where if 4...1,,, =tttt ξτη are defined in (4.4)-(4.6), then from (5.1), (5.4), (5.5), (5.6) and (5.8) we have

kkkkk PP 3211 ξγτγηγ ++≤−

∞

+ ,

where 3,2,1,max ,,,=Δ= tk

tjiji

ktγ .

Lemma 5.6. Given initial control polygon ,,0, jiji pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (4.1) together with (4.2). If ,,

kjip 0≥k

1<ψ , then

( ) ,3,2,1,0 =≤ tkkkt γψγ (5.9)

where ψ and are defined in (4.3) and (4.8). ktγ

Proof: Using (4.1), (4.2) and by utilizing the approach used in the derivation of (3.7) and (3.8) we get

( )∑ ∑= =

−++

−−−+ ⎟

⎠

⎞⎜⎝

⎛−=−

m

s

m

r

ksjri

kr

kr

ks

kji

kji pabapp

0 0

1,

1112,22,12 ( )∑ ∑

=

−

=

−++

−+++

−− ⎟⎠

⎞⎜⎝

⎛−=

m

s

m

r

ksjri

ksjri

kr

ks ppCa

0

1

0

1,

1,1

11 , (5.10)

where , ( )∑=

−−− −=r

t

kt

kt

kr baC

0

111

(∑ ∑= =

−++

−+++

−−++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

s

m

r

ksjri

ksjri

kr

ks

kji

kji ppDapp

0 0

1,

1,1

112,122,22 )

)

)

)

)

)

)

)

, (5.11)

where ,111 −−− −= kr

kr

kr CaD

(∑ ∑=

−

=

−++

−+++

−−+++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

s

m

r

ksjri

ksjri

kr

ks

kji

kji ppCbpp

0

1

0

1,

1,1

1112,212,12 , (5.12)

(∑ ∑= =

−++

−+++

−−++++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

s

m

r

ksjri

ksjri

kr

ks

kji

kji ppDbpp

0 0

1,

1,1

1112,1212,22 , (5.13)

(∑ ∑=

−

=

−+++

−++++

−−+++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

s

m

r

ksjri

ksjri

kr

ks

kji

kji ppCapp

0

1

0

11,

11,1

1122,222,12 , (5.14)

(∑ ∑= =

−+++

−++++

−−++++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

s

m

r

ksjri

ksjri

kr

ks

kji

kji ppDapp

0 0

11,

11,1

1122,1222,22 , (5.15)

(∑ ∑=

−

=

−++

−+++

−−+ ⎟

⎠

⎞⎜⎝

⎛−=−

m

r

m

s

ksjri

ksjri

ks

kr

kji

kji ppCapp

0

1

0

1,

11,

112,212,2 , (5.16)

(∑ ∑= =

−++

−+++

−−++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

r

m

s

ksjri

ksjri

ks

kr

kji

kji ppDapp

0 0

1,

11,

1112,222,2 , (5.17)

(∑ ∑=

−

=

−++

−+++

−−+++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

r

m

s

ksjri

ksjri

ks

kr

kji

kji ppCbpp

0

1

0

1,

11,

112,1212,12 , (5.18)

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Journal of Information and Computing Science, 2 (2007) 3, pp 179-190 189

)(∑ ∑=

−

=

−++

−+++

−−++++ ⎟

⎠

⎞⎜⎝

⎛−=−

m

r

m

s

ksjri

ksjri

ks

kr

kji

kji ppDbpp

0

1

0

1,

11,

1112,1222,12 . (5.19)

Using (5.10)-(5.13) we get

11,,,

1

0

1

0

11,,,

maxmax −−

=

−

=

− Δ⎟⎠

⎞⎜⎝

⎛≤Δ ∑∑ k

jiji

m

s

ks

m

r

kr

kjiji

Ca ,

11,,,0

1

0

11,,,

maxmax −

=

−

=

− Δ⎟⎠

⎞⎜⎝

⎛≤Δ ∑∑ k

jiji

m

s

ks

m

r

kr

kjiji

Da ,

11,,,

1

0

1

0

11,,,

maxmax −−

=

−

=

− Δ⎟⎠

⎞⎜⎝

⎛≤Δ ∑∑ k

jiji

m

s

ks

m

r

kr

kjiji

Cb ,

11,,,0

1

0

11,,,

maxmax −

=

−

=

− Δ⎟⎠

⎞⎜⎝

⎛≤Δ ∑∑ k

jiji

m

s

ks

m

r

kr

kjiji

Db .

Recursively we get

01

1

0

0

0

01

0

2

0

21

0

1

0

11 ... γγ ⎟

⎠

⎞⎜⎝

⎛≤ ∑∑∑∑∑∑

−

==

−

=

−

=

−−

=

−

=

−m

ss

m

rr

m

s

ks

m

r

kr

m

s

ks

m

r

kr

k CaCaCa ,

01

0

0

0

0

0

2

0

2

0

1

0

11 ... γγ ⎟

⎠

⎞⎜⎝

⎛≤ ∑∑∑∑∑∑

===

−

=

−

=

−

=

−m

ss

m

rr

m

s

ks

m

r

kr

m

s

ks

m

r

kr

k DaDaDa ,

01

1

0

0

0

01

0

2

0

21

0

1

0

11 ... γγ ⎟

⎠

⎞⎜⎝

⎛≤ ∑∑∑∑∑∑

−

==

−

=

−

=

−−

=

−

=

−m

ss

m

rr

m

s

ks

m

r

kr

m

s

ks

m

r

kr

k CbCbCb ,

01

0

0

0

0

0

2

0

2

0

1

0

11 ... γγ ⎟

⎠

⎞⎜⎝

⎛≤ ∑∑∑∑∑∑

===

−

=

−

=

−

=

−m

ss

m

rr

m

s

ks

m

r

kr

m

s

ks

m

r

kr

k DbDbDb ,

where kjiji

k1,,,1 max Δ=γ .

From above inequalities and (4.3) we get ( ) 011

ˆˆ γγkk CA≤ , ( ) 0

11ˆˆ γγ

kk DA≤ , ,

, where are defined in (4.3).

( ) 011

ˆˆ γγkk CB≤

( ) 011

ˆˆ γγkk DB≤ DCBA ˆ,ˆ,ˆ,ˆ

If { }DBCBDACA ˆˆ,ˆˆ,ˆˆ,ˆˆmax=ψ , then we get ( ) 011 γψγ kk ≤ . Using (5.14) and (5.15) recursively we get

. Similarly from (5.16)-(5.20) we get ( ) 033 γψγ kk ≤ ( ) 0

22 γψγ kk ≤ . This completes the proof.

Remark 5.1. The main problem in Theorem 2 [3] is that its proof contains errors on page 609 in Equations (38), (40), (42), (44) and (46). Their corrected forms are given in Equations (5.10), (5.12), (5.14), (5.16) and (5.18) respectively of our Lemma 5.6.

Theorem 5.7. Given initial control polygon ,,0, jiji pp = ,Zi∈ let the values be defined

recursively by non-stationary subdivision scheme (4.1) together with (4.2). Suppose

,,k

jip 0≥kkP be the piecewise

linear interpolation to the values and kjip ,

∞P be the limit surface of the subdivision scheme (4.1). If 1<ψ , then error bounds between limit surface and its control polygon after k-fold subdivision is

{ } ( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

++≤−∞

∞

ψψξγτγηγ

103

02

01

kk PP , (5.20)

where { }4321 ,,,max ηηηηη = , { }4321 ,,,max τττττ = and { }4321 ,,,max ξξξξξ = , 4...1,,, =tttt ξτη are

defined in (4.4)-(4.6), are defined in (4.8) and ,3,2,1,0 =ttγ ψ is defined in (4.3).

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G. Mustafa, et al: Estimating error bounds for non-stationary binary subdivision curves/surfaces 190

Proof: From (5.7) we have kkkkk PP 3211 ξγτγηγ ++≤−

∞

+ . Using (5.9) we get

( )( )kkk PP ψξγτγηγ 03

02

01

1 ++≤−∞

+ .

By triangular inequality { } ( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

++≤−∞

∞

ψψξγτγηγ

103

02

01

kk PP . This completes the proof.

Remarks 5.2. Theorem 5.7 is a generalization of the corresponding Theorem 7 (after correction) of Mustafa et al. In particular, if the subdivision mask does't change by increasing subdivision level k then both results coincides.

By using Theorem 5.7 we can obtain the following computational formula of subdivision depth for non-stationary binary subdivision surfaces.

Theorem 5.8. Let be subdivision depth and let be the error bound between non-stationary binary

subdivision surface and its -level control mesh

k kd

k kP . For arbitrary 0>ε , if ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−++

≥ −

ψεξγτγηγ

ψ 1log

03

02

01

1k

then . ε≤kd

Proof: From (5.20), we have { } ( )⎟⎟⎠

⎞⎜⎜⎝

⎛−

++≤−=∞

∞

ψψξγτγηγ

103

02

01

kkk PPd , This implies, for

arbitrary given 0>ε , when subdivision depth satisfy the following inequality k

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−++

≥ −

ψεξγτγηγ

ψ 1log

03

02

01

1k

then . This completes the proof. ε≤kd

6. References [1] F. Cheng, Estimating subdivision depths for rational curves and surfaces, ACM Transactions on Graphics. 1992,

11(2): 140-151. [2] F. Cheng and J. H. Yong. Subdivision depth computation for Catmull-Clark subdivision surface. Computer Aided

Geometric Design and Applications. 2006, 3(1-4): 485-494. [3] Ghulam Mustafa, Chen Falai, and Jiansong Deng. Estimating error bounds for binary subdivision curves/surfaces.

J. Comp. Appl. Math. 2006, 193: 596-613. [4] M. K. Jena, P. Shunmugaraj and P. C. Das. A non-stationary subdivision scheme for curve interpolation. ANZIAM

J. 2003, 44(E): 216-235. [5] M. I. Karavelas, P. D. Kaklis and K. V. Kostas. Bounding the distance between 2D parametric Bézier curves and

their control polygon. Computing. 2004, 72 (72): 117-128. [6] D. Lutterkort and J. Peters. Tight linear envelopes for splines. Numer. Math. 2001, 89: 735-748. [7] D. Nairn, J. Peters and D. Lutterkort. Sharp quantitative bounds on the distance between a polynomial piece and

its Bézier control polygon. Computer Aided Geometric Design. 1999, 16: 613-631. [8] Wang Huawei, Guan Youjiang and Qin Kaihuai. Error estimate for Doo-Sabin surfaces. Progress in Natural Sci.

2002, 12(9): 697-700. [9] Wang Huawei and Qin Kaihuai. Estimating subdivision depth of Catmull-Clark surfaces. J. Computer Sci and

Tech. 2004, 19(5): 657-664. [10] Xiao-Ming Zeng and X. J. Chen. Computational formula of depth for Catmull-Clark subdivion surfaces. J. Comp.

Appl. Math. 2006, 195(1-2): 252-262.

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